热流科学与工程西步文源大堂G教育部重点实验室3.1.5 Only fully implicit scheme can guaranteephysically meaningful solutionIllustrated by an example/[Known] 1-D transient HC withoutTinitialsource term, uniform initialfield. Two surfaces weresuddenly cooled downtozero.[Find]Variation of inner point0xtemperature with time-W[Solution] Discretized by Practice AFAdopting three grids: W, P, and EAxAxPhysically the variation trendshown in right fig. can be expected!ΦHFO-NHTCE16/55CENTER
16/55 3.1.5 Only fully implicit scheme can guarantee physically meaningful solution Illustrated by an example. 1-D transient HC without source term, uniform initial field. Two surfaces were suddenly cooled down to zero. [Known] Variation of inner point temperature with time [Find] [Solution] Discretized by Practice A Adopting three grids: W, P, and E. Physically the variation trend shown in right fig. can be expected!
热流科学与工程西步文源大堂E教育部重点实验室Analyzing the 2nd time level:Substituting:T,=T=Tw=T=0;Sc=0, S,=0apT,=aLfTe+(1-f)TI+awLfTw+(1-f)TaT[a, -(1- f)ae-(1-f)aw +(1- f)8,A,(x)Ax)+8cA,(x)AxYieldsa,T, =T,[a,-(1- f)ae-(1-f)aw]a-(1-f)(aw+ae)_ap-(1-f)(aw+ag)Tpi.e.:Toa + f(aw +ae)apa/△xat△tAx2a.1agpcn2a.Ar?Ar?pc,Ax/ △tAxr△tpcas1-2(1- f)a\tT.Grid FourierFoFinally4ToAr2number!aAt1+2f中CFD-NHT-EHT17/55ACENTER
17/55 Analyzing the 2nd time level: 0 0 0 ; T T T T E E W W Yields 0 0 [ (1 ) (1 ) ] P P E W P P a T T f a f a a 0, 0 C P S S i.e.： 0 0 0 0 (1 )( ) (1 )( ) ( ) P P W E P W E P P P W E T a f a a a f a a T a a f a a 1 , E W a a x 0 , p P c x a t Finally: 0 2 2 1 2(1 )( ) 1 2 ( ) P P a t x a f T T t x f 0 2 2 / ( ) / E P p p a x t a t a c x t c x x 0 0 [ (1 ) ] [ (1 ) ] P P E E E W W W a T a fT f T a fT f T 0 0 [ (1 ) (1 ) (1 ) ( ) ] ( ) T f a f a f S A x x S A P P E W P P C P a x x Substituting： 0 0 0 0 0 0 2 a t Fo x Grid Fourier number!
热流科学与工程西步文通大堂E教育部重点实验室Tp1-2(1- f)Fo1.0Tp1+2 fFoOnly fully implicit0.8scheme can guaranteePhysically it ispositive ratio0.6required :全隐格式（f=1）0.4T.0.2>0To0.02412142×F0AT-0.275二initial0.5)Crauk-NColson(=-0.4-0.6..显式(=-0)-0.8041.0WEAxAxCFD-NHT-EHTG18/55CENTER
18/55 0 1 2(1 ) 1 2 P P T f Fo T fFo Physically it is required ： 0 0 P P T T Only fully implicit scheme can guarantee positive ratio
热流科学与工程西步文源大堂G教育部重点实验室Only when f =1 (fully imp.) can guarantee it!This result can beobtained from physical精确解analysis!Crank-NicolsonThe discretized form全隐格式Rof transient HC is:XDapT,=aT +awT +a,T, +b ()。Mphysically all coefficients0must by ≥ 0 :2Oscillating boundarya, =a-(1-f)ae-(1-f)aw≥0heat flux1-(1-f)(ae +aw)/a, ≥00510201510°at/L21adtae=FoFoa.Ar?2(1- f)中CED-NHT-EHT19/55CENTER
19/55 Only when f 1 (fully imp.）can guarantee it! This result can be obtained from physical analysis! 0 (1 ) (1 ) 0 t P E W a a f a f a 0 1 (1 )( ) / 0 E W P f a a a 1 2(1 ) Fo f physically all coefficients must by 0 : The discretized form of transient HC is: 0 2 = E P a a t Fo a x Oscillating boundary heat flux f f 0 P P E E W W t P a T a T a T a T b